AQA A Level Maths June 2023 Paper 3: question paper and mark scheme (7357/3)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the AQA paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Algebra and functions · 7 marks

   Express ${\displaystyle \frac{x^3+1}{x^2-1}}$ in the form $Ax+B+{\displaystyle \frac{C}{x-1}}+{\displaystyle \frac{D}{x+1}}$ and find $A,B,C,D$.

   Worked solution

   Polynomial division: $\frac{x^3+1}{x^2-1}=x+\frac{x+1}{x^2-1}=x+\frac{1}{x-1}$ (since $\frac{x+1}{(x+1)(x-1)}=\frac{1}{x-1}$)   M1 M1 A1. So $A=1$, $B=0$, $C=1$, $D=0$   A1 A1 A1 A1.

2. Exponentials and logarithms · 5 marks

   The variables $x$ and $y$ satisfy the equation $y=a{b}^x$. A graph of $\ln y$ against $x$ is plotted and forms a straight line passing through $(0,2)$ and $(4,10)$. Find the exact values of the constants $a$ and $b$.

   Worked solution

   1. M1 for taking natural logarithms of both sides to get $\ln y=\ln a+x\ln b$.
   2. M1 for calculating the gradient of the line: $\ln b=\frac{10-2}{4-0}=2$.
   3. A1 for solving to find exactly $b=e^2$.
   4. M1 for identifying the y-intercept from the coordinate $(0,2)$, so $\ln a=2$.
   5. A1 for solving to find exactly $a=e^2$.

3. Sampling and data · 5 marks

   For a sample of $n=10$ observations, $\mathrm{\Sigma }x=120$ and $\mathrm{\Sigma }{x}^2=1800$. A new variable $y$ is defined as $y=3x-2$. Calculate the mean and variance of $y$.

   Worked solution

   1. M1 for calculating $\overline{x}=\frac{120}{10}=12$.
   2. M1 for finding $\overline{y}=3(12)-2=34$.
   3. M1 for calculating $Var(x)=\frac{1800}{10}-{12}^2=180-144=36$.
   4. M1 for applying the variance transformation rule $Var(y)=3^2\times Var(x)$.
   5. A1 for finding $Var(y)=9\times 36=324$.

More on these topics: Algebra and functions · Exponentials and logarithms · Sampling and data

What this paper covers

What this paper is like

• Shape: 17 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 14 marks.
• Biggest question: Q7, 14 marks: logo made of sectors and a rhombus.
• Marks by main topic: Statistical distributions 27, Differentiation 16, Trigonometry 14, Algebra and functions 12, Sampling and data 12, Integration 7, Hypothesis testing 6, Probability 5, Numerical methods 1.
• More Trigonometry in other sittings: June 2022 Q5 (3 marks), November 2021 Q5 (13 marks), November 2020 Q9 (5 marks).
• Against the other 5 sittings we mapped: more marks on Differentiation (16 marks here, 7 on average) and Trigonometry (14 marks here, 7 on average); fewer on Sequences and series (0 marks here, 7 on average) and Exponentials and logarithms (0 marks here, 7 on average).
• Not on this paper: Proof, Coordinate geometry, Sequences and series, Exponentials and logarithms, Vectors.
• Calculator: allowed.

7357/3: question by question

Labels and topics are ours, read from the 7357/3 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Equation of a graph (multiple choice) (1 marks) · Algebra and functions
2. Q2Over- and underestimates from the trapezium rule (1 marks) · Numerical methods
3. Q3Tangent at a stationary point (multiple choice) (1 marks) · Differentiation
4. Q4Writing a root expression as powers of x (2 marks) · Algebra and functions
5. Q5Gradient of an exponential curve (3 marks) · Differentiation, Exponentials and logarithms
6. Q6Cubic sketch and the factor theorem (9 marks) · Algebra and functions
7. Q7Logo made of sectors and a rhombus (14 marks) · Trigonometry
8. Q8Integration by substitution (7 marks) · Integration
9. Q9Water slide modelled by a parametric curve (12 marks) · Differentiation, Coordinate geometry
10. Q10Impossible correlation coefficient (multiple choice) (1 marks) · Sampling and data
11. Q11Mutually exclusive events (tick one) (1 marks) · Probability
12. Q12Driving test passes: binomial and a test (8 marks) · Statistical distributions, Hypothesis testing
13. Q13Coins in a money box: tree diagram (4 marks) · Probability
14. Q14Normal model for recycled cans (10 marks) · Statistical distributions
15. Q15Large data set: masses of eight cars (11 marks) · Sampling and data
16. Q16Normal model for apple diameters (9 marks) · Statistical distributions
17. Q17Testing new businesses' profits (6 marks) · Hypothesis testing

Some Paper 3 statistics questions use AQA's large data set. We don't write large data set questions, so use your class materials for those.